Equivalence of Models of Cake-Cutting Protocols

TL;DR

This paper introduces a simpler game-theoretic model for cake-cutting protocols, proves its equivalence to the existing GCC model, and explores transformations and envy-related equivalences within these frameworks.

Contribution

It presents an alternative, simpler extensive-form game model for cake-cutting protocols and proves its equivalence to the GCC model in expressive power.

Findings

BC protocols are invariant under certain modifications.

Any BC protocol can be transformed into a form where agents cut first, then choose.

BC and GCC protocols are equivalent in representing cake-cutting procedures.

Abstract

The cake-cutting problem involves dividing a heterogeneous, divisible resource fairly between $n$ agents. Br\^{a}nzei et al. [6] introduced {\em generalised cut and choose} (GCC) protocols, a formal model for representing cake-cutting protocols as trees with "cut" and "choose" nodes corresponding to the agents' actions, and if-else statements. In this paper, we identify an alternative and simpler extensive-form game model for cake-cutting protocols, that we call {\em branch choice} (BC) protocols. We show that the class of protocols we can represent using this model is invariant under certain modifications to its definition. We further prove that any such protocol can be converted to a restricted form in which the agents first cut the cake and then get to choose between various branches leading to different allocations. Finally, we show that this model has the same expressive power as GCC protocols, i.e. they represent the same class of protocols up to a notion of equivalence involving the bounds on envy that each agent can guarantee for themselves. For this purpose, we introduce a new notion of envy-equivalence of protocols.